Abstract

This paper reviews the universal critical properties exhibited by some 1d discrete dynamical systems: period-doubling systems and systems generating diffusive motion. While the period-doubling bifurcations have the universal asymptotic bifurcation rate δ=4.6692..., the tangent bifurcations present within the chaotic region do not follow this rate. We show that the tangent bifurcations giving rise to a fine structure of periodic windows have bifurcation rates γk which can be calculated analytically. They converge to a universal constant γ=2.94805... We have found that a class of dynamical systems show the onset of a diffusive motion in addition to period-doubling. The diffusion is self-generated and does not rely on the presence of random external forces. The onset of diffusion has strong analogies with a phase-transition. The diffusion coefficient is the order parameter and has a universal critical exponent. The dependence on random external fluctuations is also universal and can be expressed in terms of a universal scaling function which is calculated analytically.